Alvin E. Roth and Marilda A. Oliveira Sotomayor (1990)
Two-Sided Matching – A Study
in Game-Theoretic Modeling and Analysis
, Cambridge University Press.
L. J. Savage (1954)
The Foundations of Statistics
, John Wiley & Sons, New York.
Martin Shubik (1982)
Game Theory in the Social Sciences
, The MIT Press.
John Maynard Smith (1982)
Evolution and the Theory of Games
, Cambridge University
Press.
7

Sylvain Sorin (2002)
A First Course on Zero-Sum Repeated Games
, Math´ematiques &
Applications
37
, Springer.
Philip D. Stra
ﬃ
n (1993)
Game Theory and Strategy
, Mathematical Association of Amer-
ica.
Alan D. Taylor (1995)
Mathematics and Politics — Strategy, Voting, Power and Proof
,
Springer-Verlag, New York.
Stef Tijs (2003)
Introduction to Game Theory
, Hindustan Book Agency, India.
J. von Neumann and O. Morgenstern (1944)
The Theory of Games and Economic Behavior
,
Princeton University Press.
John D. Williams, (1966)
The Compleat Strategyst
, 2nd Edition, McGraw-Hill, New York.
8

GAME THEORY
Thomas S. Ferguson
Part I. Impartial Combinatorial Games
1. Take-Away Games.
1.1 A Simple Take-Away Game.
1.2 What is a Combinatorial Game?
1.3 P-positions, N-positions.
1.4 Subtraction Games.
1.5 Exercises.
2. The Game of Nim.
2.1 Preliminary Analysis.
2.2 Nim-Sum.
2.3 Nim With a Larger Number of Piles.
2.4 Proof of Bouton’s Theorem.
2.5 Mis`
ere Nim.
2.6 Exercises.
3. Graph Games.
3.1 Games Played on Directed Graphs.
3.2 The Sprague-Grundy Function.
3.3 Examples.
3.4 The Sprague-Grundy Function on More General Graphs.
3.5 Exercises.
4. Sums of Combinatorial Games.
4.1 The Sum of
n
Graph Games.
4.2 The Sprague Grundy Theorem.
4.3 Applications.
I – 1

4.4 Take-and-Break Games.
4.5 Exercises.
5. Coin Turning Games.
5.1 Examples.
5.2 Two-dimensional Coin Turning Games.
5.3 Nim Multiplication.
5.4 Tartan Games.
5.5 Exercises.
6. Green Hackenbush.
6.1 Bamboo Stalks.
6.2 Green Hackenbush on Trees.
6.3 Green Hackenbush on General Rooted Graphs.
6.4 Exercises.
References.
I – 2

Part I. Impartial Combinatorial Games
1. Take-Away Games.
Combinatorial games are two-person games with perfect information and no chance
moves, and with a win-or-lose outcome. Such a game is determined by a set of positions,
including an initial position, and the player whose turn it is to move. Play moves from one
position to another, with the players usually alternating moves, until a terminal position
is reached. A terminal position is one from which no moves are possible. Then one of the
players is declared the winner and the other the loser.
There are two main references for the material on combinatorial games. One is the
research book,
On Numbers and Games
by J. H. Conway, Academic Press, 1976.
This
book introduced many of the basic ideas of the subject and led to a rapid growth of the
area that continues today.
The other reference, more appropriate for this class, is the
two-volume book,
Winning Ways for your mathematical plays
by Berlekamp, Conway and
Guy, Academic Press, 1982, in paperback. There are many interesting games described in
this book and much of it is accessible to the undergraduate mathematics student. This
theory may be divided into two parts,
impartial games
in which the set of moves available